modeling valued networks with statnet

Modeling valued networks with statnet

Pavel N. Krivitsky, Carter T. Butts,The Statnet Development Team

May 22, 2013

Contents

1 Getting the software 2

2 network and edge attributes 2

2.1 Constructing valued networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Sampson’s Monks, pooled . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Zachary’s Karate club . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Visualizing a valued network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Modeling dyad-dependent interaction counts with valued ERGMs usingergm.count 8

3.1 Valued ERGMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 New concept: a reference distribution . . . . . . . . . . . . . . . . . . . 8

3.2 Initial values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Example: Sampson’s Monks with Binomial(3) reference . . . . . . . . . 14

3.2.2 Example: Zachary’s Karate Club with Poisson reference . . . . . . . . . 14

3.3 ERGM terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 GLM-style terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.2 Sparsity and zero-modification . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.3 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.4 Mutuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.5 Individual heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.6 Triadic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.1 Sampson’s Monks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.2 Zachary’s Karate Club . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Other notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Medium-term Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1

4 Latent space models with non-binary response with latentnet 26

4.1 A very quick overview of latentnet . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Specifying non-binary models in ergmm . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Medium-term Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1 Getting the software

If you have not already done so, please make sure that you have a reasonably new versionof R, preferably the latest (3.0.0) (R Core Team, 2013). Then, download and install thelatest versions of the Statnet (Handcock etãl., 2008; Goodreau etãl., 2008) packages, inparticular ergm version 3.1 (Hunter etãl., 2008; Handcock etãl., 2013), ergm.count version3.1 (Krivitsky, 2013), latentnet version 2.4-4 (Krivitsky and Handcock, 2013), and theirdependencies. You can accomplish this by typing:

> install.packages("ergm.count")

> install.packages("latentnet")

> library(ergm.count)

> library(latentnet)

We will also need a temporary workaround for a bug:

> as.mcmc.default <- coda:::as.mcmc.default

> as.mcmc.list.default <- coda:::as.mcmc.list.default

2 network and edge attributes

network (Butts, 2008; Butts etãl., March 15, 2013) objects have three types of attributes:

network attributes attributes which pertain to the whole network and include such infor-mation as network size, directedness, and multiplicity;

vertex attributes attributes which pertain to the individual vertices in the network andinclude such information as vertex label, as well as group assignment or some otherproperty of the individual being represented;

edge attributes attributes which pertain to edges in the network and include such informa-tion as edge value.

An edge attribute is only defined for edges that exist in the network. Thus, in a matterof speaking, to set an edge value, one first has to create an edge and then set its attribute.

As with network and vertex attributes, edge attributes that have been set can be listedwith list.edge.attributes. Every network has at least one edge attribute: "na", which, ifset to TRUE, marks an edge as missing.

2

2.1 Constructing valued networks

2.1.1 Sampson’s Monks, pooled

The first dataset that we’ll be using is the (in)famous Sampson’s monks. Dataset samplk inpackage ergm contains three (binary) networks: samplk1, samplk2, and samplk3, containingthe Monks’ top-tree friendship nominations at each of the three survey time points. We aregoing to construct a valued network that pools these nominations.

Method 1: From a sociomatrix Suppose that a valued sociomatrix is available.

> data(samplk)

> ls()

[1] "as.mcmc.default" "as.mcmc.list.default" "samplk1"

[4] "samplk2" "samplk3"

> as.matrix(samplk1)[1:5,1:5]

Ramuald Bonaventure Ambrose Berthold Peter

Ramuald 0 0 0 0 1

Bonaventure 0 0 0 0 1

Ambrose 0 1 0 0 0

Berthold 0 0 1 0 1

Peter 1 1 0 1 0

> # A sociomatrix totaling the nominations.

> samplk.tot.m<-as.matrix(samplk1)+as.matrix(samplk2)+as.matrix(samplk3)

> samplk.tot.m[1:5,1:5]


Ramuald 0 2 1 0 3


Ambrose 0 3 0 0 0

Berthold 0 1 3 0 3

Peter 1 3 0 3 0

> # Create a network where the number of nominations becomes an attribute of an edge.

> samplk.tot <- as.network(samplk.tot.m, directed=TRUE, matrix.type="a",

ignore.eval=FALSE, names.eval="nominations" # Important!

)

> # Add vertex attributes.

> samplk.tot %v% "group" <- samplk1 %v% "group" # Groups identified by Sampson

> samplk.tot %v% "group"

[1] "Waverers" "Loyal" "Loyal" "Loyal" "Loyal" "Loyal"

[7] "Waverers" "Turks" "Turks" "Turks" "Turks" "Turks"

[13] "Turks" "Turks" "Waverers" "Outcasts" "Outcasts" "Outcasts"

3

> samplk.tot %v% "vertex.names" <- samplk1 %v% "vertex.names" # Names

> # We can view the attribute as a sociomatrix.

> as.matrix(samplk.tot,attrname="nominations")[1:5,1:5]


Ramuald 0 2 1 0 3


Ambrose 0 3 0 0 0

Berthold 0 1 3 0 3

Peter 1 3 0 3 0

> # Also, note that samplk.tot now has an edge if i nominated j *at least once*.

> as.matrix(samplk.tot)[1:5,1:5]


Ramuald 0 1 1 0 1


Ambrose 0 1 0 0 0

Berthold 0 1 1 0 1

Peter 1 1 0 1 0

Method 2: From an edgelist Now, suppose that instead we have an edgelist with values:

> samplk.tot.el <- as.matrix(samplk.tot, attrname="nominations",

matrix.type="edgelist")

> samplk.tot.el[1:5,]

[,1] [,2] [,3]

[1,] 5 1 1

[2,] 7 1 1

[3,] 1 2 2

[4,] 3 2 3

[5,] 4 2 1

> # and an initial empty network.

> samplk.tot2 <- samplk1 # Copy samplk1

> delete.edges(samplk.tot2, seq_along(samplk.tot2$mel)) # Empty it out

> samplk.tot2

Network attributes:

vertices = 18

directed = TRUE

hyper = FALSE

loops = FALSE

multiple = FALSE

bipartite = FALSE

4

total edges= 0

missing edges= 0

non-missing edges= 0

Vertex attribute names:

cloisterville group vertex.names

> samplk.tot2[samplk.tot.el[,1:2], names.eval="nominations", add.edges=TRUE] <-

samplk.tot.el[,3]

> as.matrix(samplk.tot2,attrname="nominations")[1:5,1:5]


Ramuald 0 2 1 0 3


Ambrose 0 3 0 0 0

Berthold 0 1 3 0 3

Peter 1 3 0 3 0

In general, the construction net[i,j, names.eval="attrname", add.edges=TRUE] <-

value can be used to modify individual edge values for attribute "attrname". This way, wecan also add more than one edge attribute to a network.

2.1.2 Zachary’s Karate club

The other dataset we’ll be using is almost as (in)famous Zachary’s Karate Club dataset is acollapsed multiplex network that counts the number of social contexts in which each pair ofindividuals associated with the Karate Club in question interacted. A total of 8 contexts wereconsidered, but as the contexts themselves were determined by the network process, this limititself can be argued to be endogenous.

Over the course of the study, the club split into two factions one led by the instructor, “Mr.Hi” and the other led by the Club President, “John A.”. Zachary also recorded the factionalignment of every regular attendee in the club. This dataset is included in the ergm.count

package, as zach.

2.2 Visualizing a valued network

The network’s plot method for networks can be used to plot a sociogram of a network.When plotting a valued network, we might want to do color the ties depending on theirvalue. Function gray can be used to generate a gradient of colors, with gray(0) generatingblack and gray(1) generating white. This can then be passed to the edge.col argument ofplot.network.

Sampson’s Monks For the monks, let’s do it as a matrix.

5

> par(mar=rep(0,4))

> samplk.ecol <-

matrix(gray(1 - (as.matrix(samplk.tot, attrname="nominations")/3)),

nrow=network.size(samplk.tot))

> plot(samplk.tot, edge.col=samplk.ecol, usecurve=TRUE, edge.curve=0.0001,

displaylabels=TRUE, vertex.col=as.factor(samplk.tot%v%"group"))

Ramuald

Bonaventure

Ambrose

Berthold

Peter

Louis

Victor

Winfrid

John Bosco Gregory

HughBoniface

Mark

Albert

Amand

Basil Elias

Simplicius

Edge color can also be passed as a vector of colors corresponding to edges. It’s moreefficient, but the ordering in the vector must correspond to network object’s internal ordering,so it should be used with care.

Zachary’s Karate Club In the following plot, we plot those strongly aligned with Mr. Hias red, those with John A. with purple, those neutral as green, and those weakly aligned with

6

colors in between.

> data(zach)

> zach.ecol <- gray(1 - (zach %e% "contexts")/8)

> zach.vcol <- rainbow(5)[zach %v% "faction.id"+3]

> par(mar=rep(0,4))

> plot(zach, edge.col=zach.ecol, vertex.col=zach.vcol, displaylabels=TRUE)

Mr. Hi (1)

2

3 4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19 20

21

22

23

24

2526

27

2829

30

31

32

33John A. (34)

7

3 Modeling dyad-dependent interaction counts with valuedERGMs using ergm.count

Many of the functions in package ergm, including ergm, simulate, and summary, have beenextended to handle networks with valued relatins. They switch into this valued mode whenpassed the response argument, specifying the name of the edge attribute to use as the responsevariable. For example, a new valued term sum evaluates the sum of the values of all of therelations:

∑(i,j)∈Y yi,j . So,

> summary(samplk.tot~sum)

produces an error (because no such term has been implemented for binary mode), while

> summary(samplk.tot~sum, response="nominations")

sum

168

gives the summary statistics. We will introduce more statistics shortly. First, we need tointroduce the notion of valued ERGMs.

For a more in-depth discussion of the following, see (Krivitsky, 2012).

3.1 Valued ERGMs

Generally, a valued ERGM (for discrete variables) looks like this:

Prh,g(Y = y;θ) =h(y) exp(θ>g(y))

ch,g(θ), y ∈ Y,

where the normalizing constant

ch,g(θ) =∑y∈Y

h(y) exp(θ>g(y)

).

3.1.1 New concept: a reference distribution

With binary ERGMs, we only concern ourselves with presence and absence of ties amongactors — who is connected with whom? If we want to model values as well, we need tothink about who is connected with whom and how strong or intense these connections are. Inparticular, we need to think about how the values for connections we measure are distributed.The reference distribution (a reference measure, for the mathematically inclined) specifiesthe model for the data before we add any ERGM terms, and is the first step in modelingthese values. The reference distribution is specified using a one-sided formula as a reference

argument to an ergm or simulate call. Running

> help("ergm-references")

will list the choices implemented in the various packages, and are given as a one-sided formula.

Conceptually, it has two ingredients: the sample space and the baseline distribution (h(y)).An ERGM that “borrows” these from a distribution X for which we have a name is called anX-reference ERGM.

8

The sample space For binary ERGMs, the sample space Y — the set of possible networksthat can occur — is some subset of 2Y, the set of all possible ways in which relationshipsamong the actors may occur.

For the sample space of valued ERGMs, we need to define S, the set of possible values eachrelationship may take. For example, for count data, that’s S = {0, 1, . . . , s} if the maximumcount is s and {0, 1, . . . } if there is no a priori upper bound. Having specified that, Y isdefined as some subset of SY: the set of possible ways to assign to each relationship a value.

As with binary ERGMs, other constraints like degree distribution may be imposed on Y.

h(y): The baseline distribution What difference does it make?

Suppose that we have a sample space with S = {0, 1, 2, 3} (e.g., number of monk-monknominations) and let’s have one ERGM term: the sum of values of all relations:

∑(i,j)∈Y yi,j :

Prh,g(Y = y;θ) ∝ h(y) exp

θ ∑(i,j)∈Y

yi,j

.

There are two values for h(y) that might be familiar:

h(y) = 1 (or any constant) =⇒ Y i,ji.i.d.∼ Uniform or truncated geometric

h(y) =(

myi,j

)= m!yi,j !(m−yi,j)! =⇒ Y i,j

i.i.d.∼ Binomial(m, logit-1(θ))

What do they look like? Let’s simulate!

> y <- network.initialize(2,directed=FALSE) # A network with one dyad!

> ## Discrete Uniform reference

> # 0 coefficient: discrete uniform

> sim.du3<-simulate(y~sum, coef=0, reference=~DiscUnif(0,3),

response="w",statsonly=TRUE,nsim=1000)

> # Negative coefficient: truncated geometric skewed to the right

> sim.trgeo.m1<-simulate(y~sum, coef=-1, reference=~DiscUnif(0,3),


> # Positive coefficient: truncated geometric skewed to the left

> sim.trgeo.p1<-simulate(y~sum, coef=+1, reference=~DiscUnif(0,3),


> # Plot them:

> par(mfrow=c(1,3))

> hist(sim.du3,breaks=diff(range(sim.du3))*4)

> hist(sim.trgeo.m1,breaks=diff(range(sim.trgeo.m1))*4)

> hist(sim.trgeo.p1,breaks=diff(range(sim.trgeo.p1))*4)

9

Histogram of sim.du3

sim.du3

Fre

quen

cy

0.0 1.0 2.0 3.0

050

100

150

200

250

Histogram of sim.trgeo.m1

sim.trgeo.m1

Fre

quen

cy

0.0 1.0 2.0 3.0

010

030

050

0

Histogram of sim.trgeo.p1

sim.trgeo.p1

Fre

quen

cy

0.0 1.0 2.0 3.0

010

030

050

0

> ## Binomial reference

> # 0 coefficient: Binomial(3,1/2)

> sim.binom3<-simulate(y~sum, coef=0, reference=~Binomial(3),


> # -1 coefficient: Binomial(3, exp(-1)/(1+exp(-1)))

> sim.binom3.m1<-simulate(y~sum, coef=-1, reference=~Binomial(3),


> # +1 coefficient: Binomial(3, exp(1)/(1+exp(1)))

> sim.binom3.p1<-simulate(y~sum, coef=+1, reference=~Binomial(3),


> # Plot them:

> par(mfrow=c(1,3))

> hist(sim.binom3,breaks=diff(range(sim.binom3))*4)

> hist(sim.binom3.m1,breaks=diff(range(sim.binom3.m1))*4)

> hist(sim.binom3.p1,breaks=diff(range(sim.binom3.p1))*4)

10

Histogram of sim.binom3

sim.binom3

Fre

quen

cy

0.0 1.0 2.0 3.0

010

020

030

0Histogram of sim.binom3.m1

sim.binom3.m1

Fre

quen

cy

0.0 1.0 2.0 3.0

010

020

030

040

0

Histogram of sim.binom3.p1

sim.binom3.p1

Fre

quen

cy

0.0 1.0 2.0 3.0

010

020

030

040

0

Now, suppose that we don’t have an a priori upper bound on the counts — S = {0, 1, . . . }— then there are two familiar reference distributions:

h(y) = 1 (or any constant) =⇒ Y i,ji.i.d.∼ Geometric(p = 1− exp(θ))

h(y) = 1/∏

(i,j)∈Y yi,j ! =⇒ Y i,ji.i.d.∼ Poisson(µ = exp(θ))

> sim.geom<-simulate(y~sum, coef=log(2/3), reference=~Geometric,


> mean(sim.geom)

[1] 1.955

> sim.pois<-simulate(y~sum, coef=log(2), reference=~Poisson,


> mean(sim.pois)

[1] 2.024

Similar means. But, what do they look like?

> par(mfrow=c(1,2))

> hist(sim.geom,breaks=diff(range(sim.geom))*4)

> hist(sim.pois,breaks=diff(range(sim.pois))*4)

11

Histogram of sim.geom

sim.geom

Fre

quen

cy

0 2 4 6 8 12

010

025

0Histogram of sim.pois

sim.pois

Fre

quen

cy

0 2 4 6 8

010

025

0

Where did log(2) and log(2/3) come from? Later.

Warning: Parameter space What happens if we simulate from a geometric-referenceERGM with all coefficients set to 0?

> par(mfrow=c(1,1))

> sim.geo0<-simulate(y~sum, coef=0, reference=~Geometric,

response="w",statsonly=TRUE,nsim=100,

control=control.simulate(MCMC.burnin=0,MCMC.interval=1))

> mean(sim.geo0)

[1] 17925807

> plot(c(sim.geo0),xlab="MCMC iteration",ylab="Value of the tie")

12

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●

●●●●●●●●●

●

●●●●

●●●

●

●

●

●

●●

●●

●●●

●●●●●

●●●●

●●●

0 20 40 60 80 100

0e+

006e

+07

MCMC iteration

Val

ue o

f the

tie

Why does it do that? Because

Prh,g(Y = y;θ) =exp(θ

∑(i,j)∈Y yi,j)

ch,g(θ)

for θ ≥ 0, is not a valid distribution, because ch,g(θ) = ∞. Using reference=~Geometric

can be dangerous for this reason. This issue only arises with ERGMs that have an infinitesample space.

3.2 Initial values

Before we can start fitting these models, we need a place to start. Unfortunately, there is nogeneral MPLE for these models that we have (yet), so there are two options:

~0 This is the default. It can work reasonably well for some models, but it can also bog downfor larger networks.

A closed-form submodel If the model fit has a simpler submodel that we can estimateexactly, we can use that as a starting point. That is, we find a coefficient for thesubmodel, and then start the main fit with all other coefficients set to 0.

Just as almost ever application of binary ERGMs uses the edges term as the baseline,almost every model we are likely to use will have the baseline intensity (sum) term. Given anetwork y with attribute "a" to be used as the response, the following should give adequatestarting coefficients for sum for:

Bernoulli: Binary ERGM. Use MPLE.

Binomial(trials):p <- sum(y %e% "a")/trials/network.dyadcount(y)

init.sum <- log(p/(1-p))

13

Poisson:m <- sum(y %e% "a")/network.dyadcount(y)

init.sum <- log(m)

Geometric:m <- sum(y %e% "a")/network.dyadcount(y)

init.sum <- log(1-1/(m+1))

This will probably be automated in the next version.

3.2.1 Example: Sampson’s Monks with Binomial(3) reference

Suppose our model of interest contains the term sum, and we are using the Binomial(3)

reference distribution. Then, plugging in samplk.tot for y and "nominations" for "a",

> p <- sum(samplk.tot %e% "nominations")/3/network.dyadcount(samplk.tot)

> p

[1] 0.1830065

> samplk.sum.init <- log(p/(1-p)) # i.e., logit(p)

> samplk.sum.init

[1] -1.496109

3.2.2 Example: Zachary’s Karate Club with Poisson reference

Suppose our model of interest contains the term sum, and we are using the Poisson referencedistribution. Then, plugging in zach for y and "contexts" for "a",

> m <- sum(zach %e% "contexts")/network.dyadcount(zach)

> m

[1] 0.4117647

> zach.sum.init <- log(m)

> zach.sum.init

[1] -0.8873032

We will be using these in the following examples.

3.3 ERGM terms

3.3.1 GLM-style terms

Many of the familiar ERGM effects can be modeled using the very same terms, but applied alittle differently.

14

Any dyad-independent binary ERGM statistic can be expressed as gk =∑

(i,j)∈Y xk,i,jyi,jfor some covariate matrix xk. If yi,j is allowed to have values other than 0 and 1, then simplyusing such a term in a Poisson-reference ERGM creates the familiar log-linear effect. Similarly,in a Binomial-reference ERGM, such terms produce an effect on log-odds of a success.

The good news is that almost every dyad-independent ergm term has been reimplementedto allow this. It is invoked by specifying“form="sum"”argument for one of the terms inheritedfrom binary ERGMs, though this not required, as it’s the default. Also, note that for valuedERGMs, the “intercept” term is sum, not edges.

> help("ergm-terms")

has the complete list.

Example: Sampson’s Monks For example, we can fit the equivalent of logistic regressionon the probability of nomination, with every ordered pair of monks observed 3 times. We will

look at differential homophily on group. That is, Y i,jind.∼ Binomial(3,πi,j) where

logit(πi,j) = β1 + β2I (i and j are both in the Loyal Opposition)

+ β3I (i and j are both Outcasts) + β4I (i and j are both Young Turks)

+ β5I (i and j are both Waverers)

To actually fit it, we need to specify the initial parameters. How many do we need?

> npar <- length(summary(samplk.tot~sum + nodematch("group",diff=TRUE,form="sum"),

response = "nominations"))

> npar

[1] 5

> samplk.tot.nm <-

ergm(samplk.tot~sum + nodematch("group",diff=TRUE,form="sum"),

response="nominations", reference=~Binomial(3),

control=control.ergm(init=c(samplk.sum.init, rep(0, npar-1))))

> mcmc.diagnostics(samplk.tot.nm)

> summary(samplk.tot.nm)

==========================

Summary of model fit

==========================

Formula: samplk.tot ~ sum + nodematch("group", diff = TRUE, form = "sum")

Iterations: 20

15

Monte Carlo MLE Results:

Estimate Std. Error MCMC % p-value

sum -2.3253 0.1346 0 <1e-04 ***

nodematch.sum.group.Loyal 2.2553 0.2940 0 <1e-04 ***

nodematch.sum.group.Outcasts 3.5661 0.5886 0 <1e-04 ***

nodematch.sum.group.Turks 2.1999 0.2202 0 <1e-04 ***

nodematch.sum.group.Waverers 1.0732 0.5813 0 0.0658 .

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Null Deviance: 0.0 on 306 degrees of freedom

Residual Deviance: -555.5 on 301 degrees of freedom

Note that the null model likelihood and deviance are defined to be 0.

AIC: -545.5 BIC: -526.8 (Smaller is better.)

Based on this, we can say that the odds of a monk nominating another monk not inthe same group during a given time step are exp(β1) = exp(−2.3253) = 0.0978, that theodds of a Loyal Opposition monk nominating another Loyal Opposition monk are exp(β2) =exp(2.2553) = 9.5379 times higher, etc..

Example: Zachary’s Karate Club We will use a Poisson log-linear model for the numberof contexts in which each pair of individuals interacted, as a function of whether this individual

is a faction leader (Mr. Hi or John A.) That is, Y i,jind.∼ Poisson(µi,j) where

log(µi,j) = β1 + β2(I (i is a faction leader) + I (j is a faction leader))

We will do this by constructing a dummy variable, a vertex attribute "leader":

> unique(zach %v% "role")

[1] "Instructor" "Member" "President"

> # Vertex attr. "leader" is TRUE for Hi and John, FALSE for others.

> zach %v% "leader" <- zach %v% "role" != "Member"

> zach.lead <-

ergm(zach~sum + nodefactor("leader"),

response="contexts", reference=~Poisson,

control=control.ergm(init=c(zach.sum.init, 0)))

> mcmc.diagnostics(zach.lead)

> summary(zach.lead)

16

==========================


==========================

Formula: zach ~ sum + nodefactor("leader")

Iterations: 20



sum -1.2204 0.0799 0 <1e-04 ***

nodefactor.sum.leader.TRUE 1.4396 0.1176 0 <1e-04 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1





Based on this, we can say that the expected number of contexts of interaction betweentwo non-leaders is exp(β1) = exp(−1.2204) = 0.2951, that the expected number of contextsof interaction between a leader and a non-leader is exp(β2) = exp(1.4396) = 4.2192 timeshigher, and that the expected number of contexts of interaction between the two leadersis exp(2β2) = exp(2 · 1.4396) = 17.8016 times higher than that between two non-leaders.(Because the leaders were hostile to each other, this may not be a very good prediction.)

3.3.2 Sparsity and zero-modification

It is often the case that in networks of counts, the network is sparse, yet if two actors dointeract, their interaction count is relatively high. This amounts to zero-inflation.

We can model this using the binary-ERGM-based terms with the term nonzero (gk =∑(i,j)∈Y I

(yi,j 6= 0

)) and GLM-style terms with argument form="nonzero": gk =

∑(i,j)∈Y xk,i,jI

(yi,j 6= 0

).

For example,

> samplk.tot.nm.nz <-

ergm(samplk.tot~sum + nonzero + nodematch("group",diff=TRUE,form="sum"),

response="nominations", reference=~Binomial(3),

control=control.ergm(init=c(samplk.sum.init,0, 0,0,0,0)))

> mcmc.diagnostics(samplk.tot.nm.nz)

> summary(samplk.tot.nm.nz)

17

==========================


==========================

Formula: samplk.tot ~ sum + nonzero + nodematch("group", diff = TRUE,

form = "sum")

Iterations: 20



sum -0.3334 0.1968 0 0.0912 .

nonzero -2.9730 0.3220 1 <1e-04 ***

nodematch.sum.group.Loyal 1.2191 0.2220 0 <1e-04 ***

nodematch.sum.group.Outcasts 1.9709 0.4837 0 <1e-04 ***

nodematch.sum.group.Turks 1.1895 0.1779 0 <1e-04 ***

nodematch.sum.group.Waverers 0.5998 0.4195 1 0.1538

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1




AIC: -634.3 BIC: -612 (Smaller is better.)

fits a zero-modified Binomial model, with a coefficient on the number of non-zero relations−2.973 is negative and highly significant, indicating that there is an excess of zeros in thedata relative to the binomial distribution, and given the rest of the model.

3.3.3 Dispersion

Similarly, even if we may use Poisson as a starting distribution, the counts might be overdis-persed or underdispersed relative to it. For now, ergm offers two ways to do so:

Conway–Maxwell–Poisson

• Implemented by adding a CMP term to a Poisson- or geometric-reference ERGM.

• Effectively replaces the “1/y!” part of a Poisson density with “1/(y!)θCMP”.

+ Produces a continuum between a geometric distribution and a Bernoulli distribu-tion.

+ Can represent both over- and under-dispersion.

− Has the parameter space problem; also, has some theoretical issues.

18

Fractional moments

• Implemented by adding a sum(pow=1/2) term to a Poisson-reference ERGM.

• Adds a statistic of the form∑

(i,j)∈Y y1/2i,j to the model.

+ More stable.

+ For Poisson-like data,√yi,j is a variance-stabilizing transformation, so it could be

interpreted as modeling (along with sum the first two moments of√yi,j .

− Not well-understood.

− In extreme cases, creates a bimodal shape in the counts.

3.3.4 Mutuality

ergm binary mutuality statistic has the form g↔ =∑

(i,j)∈Y yi,jyj,i. It turns out that directlyplugging counts into that statistic is a bad idea. mutuality(form) is a valued ERGM term,permitting the following generalizations:

"geometric":∑

(i,j)∈Y√yi,jyj,i — can be viewed as uncentered covariance of variance-stabilized

counts

"min":∑

(i,j)∈Y minyi,j ,yj,i — easiest to interpret

"nabsdiff":∑

(i,j)∈Y−|yi,j − yj,i|

Figure 1 visualizes their effects.

3.3.5 Individual heterogeneity

Different actors may have different overall propensities to interact. This has been modeledusing random effects (as in the p2 model and using degeneracy-prone terms like k-star counts.

ergm implements a number of statistics to model it, but the one that seems to work bestso far seems to be

gactor cov.(y) =∑i∈N

1

n− 2

∑j,k∈Yi∧j<k

(√yi,j −

√y)(√yi,k −

√y),

essentially a measure of covariance between the√yi,js incident on the same actor. The term

nodesqrtcovar implements it.

3.3.6 Triadic closure

Finally, to generalize the notion of triadic closure, ergm implements very flexible transi-

tiveweights(twopath, combine, affect) and similar cyclicalweights statistics. Thetransitive weight statistic has the following general form:

gv(y) =∑

(i,j)∈Y

vaffect

(yi,j , vcombine

(v2-path(yi,k,yk,j)k∈N\{i,j}

)),

and can be “customized” by varying the three functions

19

min(yi,j,yj,i)

−∣

∣yi,j − yj,i

∣

∣

√yi,jyj,i

min(yi,j ,yj,i)

−|yi,j−yj,i|

yj,i

θ↔yj,i

00 yi,j

θ↔∆0→yi,j

i,j g↔(y)

yi,j = yj,i

Figure 1: Effect of several mutuality forms on the probability of Y i,j having a certain valuegiven a particular yj,i.

v2-path Given yi,k and yk,j , what is the strength of the two-path they form?

"min" the minimum of their values — conservative

"geomean" their geometric mean — more able to detect effects, but more likely to cause“degeneracy”

vcombine Given the strengths of the two-paths yi→k→j for all k 6= i, j, what is the combinedstrength of these two-paths between i and j?

"max" the strength of the strongest path — conservative; analogous to transitiveties

"sum" the sum of path strength — more able to detect effects, but more likely to cause“degeneracy”; analogous to triangles

vaffect Given the combined strength of the two-paths between i and j, how should they affectY i,j?

"min" conservative

"geomean" more able to detect effects, but more likely to cause “degeneracy”

These effects are analogous to mutuality.

3.4 Examples

3.4.1 Sampson’s Monks

Suppose that we want to fit a model with a zero-modified Binomial baseline, mutuality,transitive (hierarchical) triads, and cyclical (antihierarchical) triads, to this dataset. Such a

20

model has 5 parameters, so

> theta0 <- c(samplk.sum.init, 0, 0, 0 ,0)

> samplk.tot.ergm <-

ergm(samplk.tot ~ sum + nonzero + mutual("min") +

transitiveweights("min","max","min") +

cyclicalweights("min","max","min"),

reference=~Binomial(3), response="nominations",

control=control.ergm(init=theta0))

> mcmc.diagnostics(samplk.tot.ergm)

> summary(samplk.tot.ergm)

==========================


==========================

Formula: samplk.tot ~ sum + nonzero + mutual("min") + transitiveweights("min",

"max", "min") + cyclicalweights("min", "max", "min")

Iterations: 20



sum -0.04967 0.18854 2 0.7924

nonzero -3.45575 0.31520 4 <1e-04 ***

mutual.min 1.38499 0.24720 3 <1e-04 ***

transitiveweights.min.max.min 0.21051 0.16318 0 0.1980

cyclicalweights.min.max.min -0.24760 0.14486 1 0.0884 .

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Null Deviance: 0 on 306 degrees of freedom

Residual Deviance: -603 on 301 degrees of freedom


AIC: -593 BIC: -574.4 (Smaller is better.)

What does it tell us? The negative coefficient on nonzero suggests zero-inflation, there isstrong evidence of mutuality, and the positive coefficient on transitive weights and negativeon the cyclical weights suggests hierarchy, but they are not significant.

3.4.2 Zachary’s Karate Club

Now, let’s try using Poisson to model the Zachary Karate Club data: a zero-modified Pois-son, with potentially different levels of activity for the faction leaders, heterogeneity in actor

21

activity level overall, and an effect of difference in faction membership, a model that lookslike this:

> summary(zach~sum+nonzero+nodefactor("leader")+absdiffcat("faction.id")+

nodesqrtcovar(TRUE), response="contexts")

sum nonzero

231.00000 78.00000

nodefactor.sum.leader.TRUE absdiff.sum.faction.id.1

90.00000 74.00000

absdiff.sum.faction.id.2 absdiff.sum.faction.id.3

12.00000 11.00000

absdiff.sum.faction.id.4 nodesqrtcovar.centered

10.00000 16.17248

The parameter vector will have length 8, so

> theta0 <- c(zach.sum.init, rep(0,7))

A few other notes:

• We can make sampling from zero-inflated Poisson more efficient by telling ergm thatthat’s what we expect by setting MCMC.prop.weights="0inflated" control parameter.The current version has a bug that requires the degree of inflation to be set by the user,though this will be automatic in the next release, so MCMC.prop.args=list(p0=0.5) isalso needed.

• Informally, a Poisson random variable contains more “information” than a Bernoulli ran-dom variable, which means that large changes in likelihood are not necessarily symp-tomatic of a problem. Thus, it often helps to set MCMLE.trustregion, which is normally20 to something higher.

Now, for the fit and the diagnostics:

> zach.pois <-

ergm(zach~sum+nonzero+nodefactor("leader")+absdiffcat("faction.id")+

nodesqrtcovar(TRUE),

response="contexts", reference=~Poisson,

control=control.ergm(MCMLE.trustregion=100,

init=theta0,

MCMC.prop.weights="0inflated",

MCMC.prop.args=list(p0=0.5) # Should not be necessary in the next version.

), verbose=TRUE)

> mcmc.diagnostics(zach.pois)

> summary(zach.pois)

22

==========================


==========================

Formula: zach ~ sum + nonzero + nodefactor("leader") + absdiffcat("faction.id") +

nodesqrtcovar(TRUE)

Iterations: 20



sum 0.99512 0.09250 7 < 1e-04 ***

nonzero -3.85145 0.27512 11 < 1e-04 ***

nodefactor.sum.leader.TRUE 0.20062 0.06633 1 0.00261 **

absdiff.sum.faction.id.1 -0.10837 0.07810 0 0.16582

absdiff.sum.faction.id.2 -0.66743 0.19232 1 0.00056 ***

absdiff.sum.faction.id.3 -0.88254 0.21637 1 < 1e-04 ***

absdiff.sum.faction.id.4 -1.16067 0.23913 1 < 1e-04 ***

nodesqrtcovar.centered 1.81829 0.22266 1 < 1e-04 ***

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1





What does it tell us? The negative coefficient on nonzero suggests zero-inflation, thefaction leaders clearly have more activity than others, and the more ideologically separatedtwo individuals are, the less they interact. Over and above that, there is some additionalheterogeneity in how active individuals are: if i has a lot of interaction with j, it is likely thati has more with j′. Could this mean a form of preferential attachment?

We can try seeing whether there is some friend of a friend effect above and beyond that.This can be done by fitting a model with transitivity and seeing whether the coefficient issignificant, or we can perform a simulation test. In the following

• simulate unpacks the zach.pois ERGM fit, extracting the formula, the coefficient, andthe rest of the information.

• nsim says how many networks to generate.

• statsonly=TRUE says that we only want to see the simulated statistics, not the networks.

• monitor=~transitiveweights("geomean","sum","geomean") says that in addition tothe statistics used in the fit, we want simulate to keep track of the transitive weightsstatistic.

23

We do not need to worry about degeneracy in this case, because we are not actuallyusing that statistic in the model, only “monitoring” it.

> # Simulate from model fit:

> zach.sim <-

simulate(zach.pois, monitor=~transitiveweights("geomean","sum","geomean"),

nsim = 1000, statsonly=TRUE,

control=control.simulate.ergm(

MCMC.prop.weights="0inflated",

MCMC.prop.args=list(p0=0.5) # Should not be necessary in the next version.

))

> # What have we simulated?

> colnames(zach.sim)

[1] "sum"

[2] "nonzero"

[3] "nodefactor.sum.leader.TRUE"

[4] "absdiff.sum.faction.id.1"




[8] "nodesqrtcovar.centered"

[9] "transitiveweights.geomean.sum.geomean"

> # How high is the transitiveweights statistic in the observed network?

> zach.obs <- summary(zach ~ transitiveweights("geomean","sum","geomean"),

response="contexts")

> zach.obs

transitiveweights.geomean.sum.geomean

288.9793

Let’s plot the density of the simulated values of transitive weights statistic:

> par(mar=c(5, 4, 4, 2) + 0.1)

> # 9th col. = transitiveweights

> plot(density(zach.sim[,9]))

> abline(v=zach.obs)

24

100 200 300 400 500

0.00

00.

004

density.default(x = zach.sim[, 9])

N = 1000 Bandwidth = 15.16

Den

sity

> # Where does the observed value lie in the simulated?

> # This is a p-value for the Monte-Carlo test:

> min(mean(zach.sim[,9]>zach.obs), mean(zach.sim[,9]<zach.obs))*2

[1] 0.916

Looks like individual heterogeneity and faction alignment account for appearance of triadiceffects. (Notably, the factions themselves may be endogenous, if social influence is a factor.Untangling selection from influence is hard enough when dynamic network data are available.We cannot do it here.)

3.5 Other notes

• Missing (NA) edges are handled automatically for valued ERGMs as well.

• ergm has an argument eval.loglik, which is TRUE by default. For valued ERGMs,it’s quite a bit less efficient than for binary, at least for now. So, unless you need theAICs or BICs to compare models, and especially if your networks are not small, passeval.loglik=FALSE.

• Writing user terms is possible, but the API is a little different from that of binary changestatistics API.

3.6 Medium-term Roadmap

• A public release for rank ERGM implementation.

• Automatic starting values for valued ERGMs.

• More efficient sampling for valued ERGMs.

25

• Goodness of fit diagnostics.

• Other reference distributions, including ranks and continuous data. (Continuous uni-form is already implemented, but not well-understood.)

4 Latent space models with non-binary response with latent-

net

latentnet, as the name suggests, is designed to fit latent space models, but it can fit otherdyad-independent network models as well. Let

Y be the random network being modeled;

y be the observed network;

x is a p× n× n array of dyadic covariates, with

x·,i,j being a p-vector of covariates for dyad (i, j);

β be the p-vector of covariate coefficients;

Z be the n× d array of latent positions, with

Zi being the d-vector position of actor i;

δ be the n-vector of sender effects, with

δi being the sender effect of δ; and

γ being a n-vector of receiver effects, with

γi being the receiver effect of γ.

For brevity, let θ = (β,Z, δ,γ) and let θi,j = (β,Zi,Zj , δi,γj). Generally, a latent spacemodel that can be fit by latentnet has the following form:

Pr(Y = y|θ,x) =∏

(i,j)∈Y

Pr(Y i,j = yi,j |θi,j ,x·,i,j), (1)

Pr(Y i,j = yi,j |θi,j ,x·,i,j) = f(yi,j |µi,j), (2)

µi,j = g−1(ηi,j), (3)

ηi,j = x>·,i,jβ + d(Zi,Zj) + δi + γj , (4)

for some latent position effect d(·, ·). Effects supported by latentnet are Euclidean, havingd(Zi,Zj) = −|Zi − Zj | and bilinear, having d(Zi,Zj) = Z>i Zj . Latent space positions Zcan further be modeled as a Gaussian mixture, and δ and γ are likewise modeled as randomeffects.

In words, the values of individual relations are assumed to be independent (1), and eachpotential relation has a value modeled by a GLM, having some distribution with density f ,

26

parametrized by its expected value (2), which is, in turn, a function, via a GLM link functiong, of the linear predictor ηi,j (3), which is a linear function combining all the parameters (4).

A latent space model for a binary network has

f(yi,j |µi,j) = (µi,j)yi,j (1− µi,j)

1−yi,j

g(µi,j) = logit(µi,j),

a Bernoulli model with a logit link: a logistic regression model.

This is easily extended to any GLM. In particular,

Poisson log-linear model:

f(yi,j |µi,j) = exp(−µi,j

)µyi,ji,j /yi,j !

g(µi,j) = log(µi,j)

Binomial logit with m trials:

f(yi,j |µi,j) =

(m

yi,j

)(µi,j/m)yi,j (1− µi,j/m)m−yi,j

g(µi,j) = logit(µi,j)

Normal linear with variance σ2:

f(yi,j |µi,j) =1√

2πσ2exp

(−

(yi,j − µi,j)2

2σ2

)g(µi,j) = µi,j

These are currently supported by latentnet. Unfortunately, σ2 cannot be estimated bylatentnet at this time.

4.1 A very quick overview of latentnet

The workhorse function ergmm (for Exponential Random Graph Mixed Model) has a syntaxvery similar to ergm: the model is specified as network ~ term1 + term2 + ... and someof the dyad-independent ergm terms still work. (We are working on replicating the full setof ergm terms.) For a full list of terms implemented, see ? terms.ergmm. Note that ergmm,like glm and unlike ergm, sets the first covariate x1,i,j ≡ 1 for an intercept term, the (binary)ERGM equivalent of an edge count term.

For example,

> samplk.nm.l <- ergmm(samplk.tot~nodematch("group",diff=TRUE),tofit="mle", verbose=TRUE)

sets x1,i,j ≡ 1, x2,i,j ≡ I (i ∈ Loyal ∧ j ∈ Loyal), x3,i,j ≡ I (i ∈ Outcasts ∧ j ∈ Outcasts), etc.,to produce a model of the form

logit(Pr(Y i,j = 1)) = β1+β2I (i and j are both Loyal Opposition)+β3I (i and j are both Outcasts)+· · · ,

equivalent to a ergm specification of

27

> samplk.nm.e <- ergm(samplk.tot~edges+nodematch("group",diff=TRUE))

In fact, the produce the same coefficients and standard errors:

> summary(samplk.nm.l, point.est="mle", se=TRUE)

==========================


==========================

Formula: samplk.tot ~ nodematch("group", diff = TRUE)

Attribute: edges

Model: Bernoulli

MCMC sample of size 4000, draws are 10 iterations apart, after burnin of 10000 iterations.

Covariate coefficients MLE:

Estimate Std. Error z value Pr(>|z|)

edges -1.66208 0.17932 -9.2689 < 2.2e-16 ***

nodematch.group.Loyal 2.06755 0.49040 4.2161 2.486e-05 ***

nodematch.group.Outcasts 17.05930 900.30202 0.0189 0.98488

nodematch.group.Turks 2.57837 0.38577 6.6836 2.331e-11 ***

nodematch.group.Waverers 1.66208 0.83596 1.9882 0.04678 *

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

> summary(samplk.nm.e)

==========================


==========================

Formula: samplk.tot ~ edges + nodematch("group", diff = TRUE)

Iterations: 20



edges -1.6621 0.1793 NA <1e-04 ***

nodematch.group.Loyal 2.0675 0.4904 NA <1e-04 ***

nodematch.group.Outcasts 18.3450 1038.5246 NA 0.9859

nodematch.group.Turks 2.5784 0.3858 NA <1e-04 ***

nodematch.group.Waverers 1.6621 0.8360 NA 0.0477 *

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual Deviance: 289.1 on 301 degrees of freedom

AIC: 299.1 BIC: 317.7 (Smaller is better.)

28

latentnet fits latent space models using terms euclidean and bilinear, which take twomain arguments: d for dimension and G for number of clusters to use. So, for a classic exampleof the Sampson’s monks fit, i.e.,

logit(Pr(Y i,j = 1)) = β1 + |Zi −Zj |,

with Zi modeled as 3 spherical Gaussian clusters with 2 dimensions,

> samplk.d2G3<-ergmm(samplk.tot~euclidean(d=2,G=3), verbose=TRUE)

Random actor-specific effects can also be used, with terms rsender, receiver, and rso-

ciality, respectively: a receiver effects model of the form

logit(Pr(Y i,j = 1)) = β1 + |Zi −Zj |+ γj ,

can be fit with

> samplk.d2G3r<-ergmm(samplk.tot~euclidean(d=2,G=3)+rreceiver, verbose=TRUE)

> mcmc.diagnostics(samplk.d2G3r)

The results can be visualized using

> par(mfrow=c(1,2))

> # Extract a clustering

> Z.K.ref <- summary(samplk.d2G3,point.est="pmean")$pmean$Z.K

> # Plot one model, saving positions, using Z.K.ref to set reference clustering.

> Z.ref <- plot(samplk.d2G3, pie=TRUE, Z.K.ref=Z.K.ref)

> # Plot the other model, using Z.ref and Z.K.ref to ensure similar

> # orientation and coloring.

> plot(samplk.d2G3r, rand.eff="receiver", pie=TRUE, Z.ref=Z.ref, Z.K.ref=Z.K.ref)

−4 −2 0 2

−2

02

MKL Latent Positions of samplk.d2G3

Z1

Z2

samplk.tot ~ euclidean(d = 2, G = 3)

++

+

+

●

●

●

−4 −2 0 2 4

−1

13

MKL Latent Positions of samplk.d2G3r

Z1

Z2

samplk.tot ~ euclidean(d = 2, G = 3) + rreceiver

++

+

+●

●

●

All ergmm terms also take additional arguments, specifying prior distributions. Theirdefaults are generally sensible, so we will not discuss them here.

29

Table 1: GLM families implemented by latentnet

Family Link family= fam.par=list(...)

Bernoulli logit "Bernoulli"

binomial logit "binomial" trials= mPoisson log "Poisson"

Normal linear "normal" var= σ2

4.2 Specifying non-binary models in ergmm

Fitting non-binary models using ergmm requires three additional arguments:

response An edge attribute in the network whose values are to be used as the response.

family A string specifying the family and the link to be used.

fam.par A named list of parameters required by some families.

The families listed above are specified using parameter values listed in Table 1. Also, see

> ? families.ergmm

for more information.

We have our network of nomination counts, with a maximum of three “successes”, so wemight model it using a Binomial distribution, i.e.,

Pr(Y i,j = yi,j |ηi,j) =

(3

yi,j

)(logit-1(ηi,j))

yi,j (logit-1(ηi,j))3−yi,j ,

with ηi,j being the same as in the binary case.

> # Bernoulli logit fit (recall)

> # samplk.d2G3 <- ergmm(samplk.tot~euclidean(d=2,G=3))

> # Binomial(trials=3) logit fit

> samplk.ct.d2G3 <-

ergmm(samplk.tot~euclidean(d=2,G=3),response="nominations",

family="binomial",fam.par=list(trials=3), verbose=TRUE)

We can plot the fit

> # Plot them side-by-side, using edge.col argument:

> par(mfrow=c(1,2))

> plot(samplk.d2G3,pie=TRUE, Z.ref=Z.ref, Z.K.ref=Z.K.ref)

> plot(samplk.ct.d2G3,pie=TRUE, Z.ref=Z.ref, Z.K.ref=Z.K.ref, edge.col=samplk.ecol)

30

−4 −2 0 2

−2

02

MKL Latent Positions of samplk.d2G3

Z1

Z2

samplk.tot ~ euclidean(d = 2, G = 3)

++

+

+

●

●

●

−4 0 2 4

−3

02

4

MKL Latent Positions of samplk.ct.d2G3

Z1

Z2

samplk.tot ~ euclidean(d = 2, G = 3) (nominations)

++

+

+

Now, consider a latent cluster random effects model for the number of contexts of inter-actions, which takes into account that faction leaders are likely to have greater propensity to

interact than non-leaders: Y i,jind.∼ Poisson(µi,j) with

logµi,j = ηi,j = β1+β2(I (i is a faction leader)+I (j is a faction leader))+|Zi−Zj |+δi+δj .

Some ergmm are missing that (for now), but the "leader" dummy variable can be usedwith ergmm’s sendercov, receivercov, and/or socialitycov.

The term socialitycov(a) sets xk,i,j ≡ ai + aj . Its ergm name is nodecov. Now, let’s fitthe model:

> zach.d2G2S <- ergmm(zach~socialitycov("leader")+euclidean(d=2,G=2)+rsociality,

response="contexts",family="Poisson",

verbose=TRUE)

> mcmc.diagnostics(zach.d2G2S)

> summary(zach.d2G2S)

==========================


==========================

Formula: zach ~ socialitycov("leader") + euclidean(d = 2, G = 2) + rsociality

Attribute: contexts

Model: Poisson

MCMC sample of size 4000, draws are 10 iterations apart, after burnin of 10000 iterations.

Covariate coefficients posterior means:

Estimate 2.5% 97.5% Quantile of 0

edges 0.85346 0.18999 1.5176 0.00325 **

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socialitycov.leader.TRUE 1.96461 0.70017 3.3625 0.00200 **

---

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Sociality effect variance: 0.7768003.

Overall BIC: 1149.352

Likelihood BIC: 801.9636

Latent space/clustering BIC: 267.818

Sociality effect BIC: 79.57072

Covariate coefficients MKL:

Estimate

edges 0.3029993

socialitycov.leader.TRUE 2.0621726

> par(mar=c(5, 4, 4, 2) + 0.1)

> plot(zach.d2G2S, rand.eff="sociality", edge.col=zach.ecol, labels=TRUE)

32

−4 −2 0 2 4

−4

−2

02

4MKL Latent Positions of zach.d2G2S

Z1

Z2

Mr. Hi (1)

2

3

4

56

7

8

9

10

11

1213

14

1516

17

18

19

20

21

22

23

24

2526

27

28

29

30

31

32

33

John A. (34)

zach ~ socialitycov("leader") + euclidean(d = 2, G = 2) + rsociality (contexts)

+

+

+

Another useful application of socialitycov and others is when an undirected networkof counts is a product of collapsing an affiliation network of actors to events to produce anetwork of actors only, counting the number of shared events for each pair of actors. If thereis no particular pattern to the interaction, then the expected number of shared events betweeni and j would be proportional to the total number of events of i and to the number of eventsof j. If ai is the total number of events associated with actor i, then using a vertex attributeequaling to log(ai) in a socialitycov can be a good baseline model:

logµi,j = β1 + β2(log(ai) + log(aj))⇔ µi,j = exp (β1) aβ2i a

β2j ,

which produces proportionality when β2 ≈ 1. For an application of this, see Krivitsky et˜al.(2009).

33

4.3 Medium-term Roadmap

• Transition ergmm to use dyad-independent ergm terms.

• Enable latentnet to fit heterogeneous trial counts.

• Enable latentnet to fit normal distribution variance.

References

Carter˜T. Butts. network: A package for managing relational data in R. Journal of StatisticalSoftware, 24(2):1–36, May 2008. ISSN 1548-7660. URL http://www.jstatsoft.org/v24/

i02.

Carter˜T. Butts, Mark˜S. Handcock, and David˜R. Hunter. network: Classes for RelationalData. Irvine, CA, March 15, 2013. URL http://statnet.org/. R package version 1.7.2.

Steven˜M. Goodreau, Mark˜S. Handcock, David˜R. Hunter, Carter˜T. Butts, and MartinaMorris. A statnet tutorial. Journal of Statistical Software, 24(9):1–26, May 2008. ISSN1548-7660. URL http://www.jstatsoft.org/v24/i09.

Mark˜S. Handcock, David˜R. Hunter, Carter˜T. Butts, Steven˜M. Goodreau, and MartinaMorris. statnet: Software tools for the representation, visualization, analysis and simula-tion of network data. Journal of Statistical Software, 24(1):1–11, May 2008. ISSN 1548-7660.URL http://www.jstatsoft.org/v24/i01.

Mark˜S. Handcock, David˜R. Hunter, Carter˜T. Butts, Steven˜M. Goodreau, PavelÑ.Krivitsky, and Martina Morris. ergm: Fit, Simulate and Diagnose Exponential-FamilyModels for Networks. The Statnet Project (http://www.statnet.org), 2013. URLCRAN.R-project.org/package=ergm. R package version 3.1-0.

David˜R. Hunter, Mark˜S. Handcock, Carter˜T. Butts, Steven˜M. Goodreau, and MartinaMorris. ergm: A package to fit, simulate and diagnose exponential-family models for net-works. Journal of Statistical Software, 24(3):1–29, May 2008. ISSN 1548-7660. URLhttp://www.jstatsoft.org/v24/i03.

PavelÑ. Krivitsky. Exponential-family random graph models for valued networks. ElectronicJournal of Statistics, 6:1100–1128, 2012. doi: 10.1214/12-EJS696.

PavelÑ. Krivitsky. ergm.count: Fit, Simulate and Diagnose Exponential-Family Models forNetworks with Count Edges. The Statnet Project (http://www.statnet.org), 2013. URLCRAN.R-project.org/package=ergm.count. R package version 3.1-0.

PavelÑ. Krivitsky and Mark˜S. Handcock. latentnet: Latent position and cluster modelsfor statistical networks. The Statnet Project (http://www.statnet.org), 2013. URL CRAN.

R-project.org/package=latentnet. R package version 2.4-4.

PavelÑ. Krivitsky, Mark˜S. Handcock, Adrian˜E. Raftery, and Peter˜D. Hoff. Represent-ing degree distributions, clustering, and homophily in social networks with latent cluster

34

http://www.jstatsoft.org/v24/i02


http://statnet.org/



http://www.statnet.org

CRAN.R-project.org/package=ergm



CRAN.R-project.org/package=ergm.count


CRAN.R-project.org/package=latentnet

CRAN.R-project.org/package=latentnet

random effects models. Social Networks, 31(3):204–213, July 2009. ISSN 0378-8733. doi:10.1016/j.socnet.2009.04.001.

R Core Team. R: A Language and Environment for Statistical Computing. R Foundation forStatistical Computing, Vienna, Austria, 2013. URL http://www.R-project.org/.

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http://www.R-project.org/

modeling valued networks with statnet

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